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Dent, C.J. and Bialek, J.W. and Hobbs, B.F. (2011) 'Opportunity cost bidding by wind generators in forward
markets : analytical results.', IEEE transactions on power systems., 26 (3). pp. 1600-1608.
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1
Opportunity Cost Bidding by Wind Generators in
Forward Markets: Analytical Results
C.J. Dent, Member, IEEE, J.W. Bialek, Senior Member, IEEE, and B.F. Hobbs, Fellow, IEEE
Abstract—Wind generation must trade in forward electricity
markets based on imperfect forecasts of its output and real-time
prices. When the real-time price differs for generators that are
short and long, the optimal forward strategy must be based on the
opportunity costs of charges and payments in real time rather
than a central estimate of wind output. We present analytical
results for wind’s optimal forward strategy. In the risk-neutral
case, the optimal strategy is determined by the distribution of
real-time available wind capacity, and the expected real-time
prices conditioned on the forward price and wind out-turn; our
approach is simpler and more computationally efficient than
formulations requiring specification of full joint distributions or
a large set of scenarios. Informative closed-form examples are
derived for particular specifications of the wind-price dependence
structure. In the usual case of uncertain forward prices, the
optimal bidding strategy generally consists of a bid curve for
wind power, rather than a fixed quantity bid. A discussion of
the risk-averse problem is also provided. An analytical result is
available for aversion to production volume risk; however, we
doubt whether wind owners should be risk-averse with respect
to the income from a single settlement period, given the large
number of such periods in a year.
Index Terms—Power generation economics, Wind power generation, Risk analysis.
I. I NTRODUCTION
T
HE penetration of wind generation is increasing in power
systems worldwide. In contrast to conventional plant,
whose availability is mainly a matter of mechanical availability, the availability of wind generation capacity is primarily
determined by the weather.
As a consequence, the statistical properties of wind capacity
availability are very different from conventional plant, both on
planning and operating timescales. On a planning timescale,
this manifests itself as a probability distribution for available
capacity at some distant time in the future. On an operating
timescale, which will be considered here, the relevant probability distribution is that for wind out-turn conditional on
the information available when decisions are taken; a wind
generation owner cannot contract a level of output in a forward
market and be sure that this precise amount will be delivered.
When generators contract to sell power in forward markets,
imbalances between the forward-contracted volume and actual
C.J. Dent and J.W. Bialek are with the School of Engineering and
Computing Sciences, Durham University, South Road, Durham DH1 3LE,
UK (Email: [email protected], [email protected]).
B.F. Hobbs is with the Department of Geography and Environmental Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA, and
the Electricity Policy Research Group, University of Cambridge, Cambridge
CB3 9DE, UK (Email: [email protected]).
This research was funded in part by the Supergen FlexNet Consortium
(supergen-networks.org.uk), a part of the UK Research Councils’ Energy
Programme under grant EP/E04011X/1, and in part by NSF Grant ECS0621920.
output must be rectified in the real-time market. In some
systems, there is a single real-time price seen by generators
irrespective of whether they are short or long with respect
to their forward-contracted volumes [1]. In others, there are
different real-time prices for generators that are short or long.
Examples include the Great Britain [2], Scandinavian [3]
and Iberian [4] markets. Forward market bidding by wind
generation is thus a problem of optimisation under uncertainty
regarding both the production out-turn and the real-time price,
as the forward market bid must be made before precise
information on these real-time quantities is available.
Optimal trading strategies for wind generation have received
comparatively little attention in the literature. [5] introduced
the concept of optimising the forward market bid based on
imbalance costs, in both risk-neutral and risk-averse formulations. [6] then introduced a closed form expression for the optimal contract volume in the forward market given the forward
and real-time prices; the same optimisation approach was used
in [7], which compares strategies based on point prediction and
probabilistic wind forecasts, and also discusses risk-aversion.
[3] presented a stochastic mixed integer linear program (LP)
formulation, based on representing the uncertainty in wind outturn using a finite number of discrete scenarios; this has been
extended to coordinated operation of wind and hydro generation in [8]. This LP approach was extended from a two-market
system (forward and real-time) to a three-market (day-ahead,
adjustment and real-time) system in [4]; in addition, a more
efficient continuous LP model was presented, in which riskaverse preferences could be modelled. Other relevant work
includes a detailed discussion of utility metrics for risk-averse
traders [9], assessment of the cost of wind forecast errors [10]
and real-time prices [11], advanced wind forecasting methods
[12], operational strategies coordinating wind generation with
reserves, storage and hydro (e.g., [13]–[15]), and designing
power markets to minimise imbalance costs to wind generators
[16]. Relevant market modelling results have been presented
in [17] (the effect of wind forecasts on forward market prices),
and [18] (the statistical relationship between real-time and
forward prices).
This paper presents closed-form methods for optimising
a wind generation owner’s forward market bid in a twosettlement (forward and real-time) system. These require either
evaluation of a single formula, or numerical solution of a
specified integral or equation. New results are presented for
uncertain real-time prices that are correlated with the forward
price and real time (RT) wind output, and also for risk-averse
bidding strategies; for the simpler case of uncorrelated wind
out-turn and real-time prices, the more general derivation
clarifies that the expected (in the mathematical sense) real-
2
time prices should be used as the price forecast in the relevant
expressions. The benefits of analytical methods over previous
approaches requiring solution of LP models are:
•
•
•
Transparency of results. In closed form expressions and
direct formulas, it is much easier to identify what parameters drive the results obtained.
Computational efficiency. The expressions presented here
may be evaluated more quickly than LP models.
Reduced data requirements. The methods in this paper
require only specification of expectation values for realtime prices conditioned on the forward price and wind
out-turn, not full joint distributions or a scenario tree.
An analytical approach is particularly valuable when the
optimal bidding strategy model must be embedded in a larger
model, perhaps of the entire power market, or of generation
investment decisions. In contrast, it is highly undesirable
to have to embed a large stochastic LP model for wind’s
strategy in such a whole-market model. This paper provides a
computationally simpler alternative.
Section II describes the simplest case of a single realtime price. Section III then describes how multiple real-time
prices are used in some systems, and presents the optimal bid
strategy in this case. Sections IV-VI illustrate how this may be
applied, including cases with both independent and correlated
wind output and real-time prices, and also known or uncertain
forward price. In the usual case of uncertain forward prices, the
optimal bidding strategy generally consists of a bid curve for
wind power, rather than a fixed quantity bid. Finally Section
VII generalises some of the results to the case of aversion to
the risk of low real-time wind output. Throughout, examples
are based on British data; the methods may also be applied to
related market designs such as in Scandinavia [3].
II. O PTIMAL VOLUME : S INGLE R EAL -T IME P RICE
A. Forward Trading Based On Opportunity Cost
Like any other form of generation, a wind generator may
contract in the forward market to deliver power in real-time,
whether this be through a bilateral trading or pool system. It
might then deviate from its forward-contracted position, with
any resulting system imbalance being rectified in the realtime market. In contrast to conventional generation, however,
wind in real-time will almost always generate at precisely its
maximum available capacity due to its very low marginal cost
of production. Exceptions may include curtailment on system
security grounds at very high penetrations, or where there is
limited transmission capacity. This paper assumes competitive
markets in which individual wind owners are price-takers, and
wind always generates at available capacity.
Optimising wind’s strategy in the forward market is a nontrivial problem based on uncertainty in real-time wind out-turn
and power prices. The opportunity cost of a forward contract
in terms of foregone revenue in the real-time market in the
real-time market is then relevant, rather than the negligible
internal marginal cost of supply. Opportunity costs are relevant
in many aspects of power systems operations and bidding [19],
[20], such as:
Bidding in ancillary services markets based on revenue
foregone in the energy market.
• Bidding in one geographical location’s market based on
revenue foregone by not selling in other areas.
• Arbitrage between forward and real-time markets (called
”virtual bidding” in U.S. markets).
• Hydro power with a finite water resource; bids would
reflect revenue sacrificed at other times.
• Finite pollution permits for fossil fuel units.
• For peaking units, if a maximum number of starts per
year under maintenance contracts.
In all these situations, selling in one product market at one
time or place means that selling in other markets may not be
possible, and the cost of selling in the former should in part
reflect foregone revenues in the latter.
•
B. Single Real-Time Price
The simplest case of the forward contracting problem considers a wind operator selecting a contract volume q with
forward price πF (often called the system marginal price, SMP,
or market clearing price, MCP), with a single real-time (RT)
price1 Π applied to all imbalances. Π and production outturn W (conditioned on the available information at time of
contracting) are random variables with joint probability density
function fΠ,W (π, w). The expected total income is then
Z Z
πF q +
π(w − q)fΠ,W (π, w)dπdw.
(1)
Differentiating with respect to q to find the optimal bid q̂
(assuming no bounds on q̂):
Z Z
πF =
πfΠ,W (π, w)dπdw = E[Π].
(2)
This is a standard result (assuming risk-neutral behaviour by
market players) that arbitrage between the RT and forward
markets means that the forward price is the expected RT price.
Similar undramatic results may be obtained from various
generalisations of the single real-time price problem, such as
a market where a number of identical wind owners bid simultaneously, and where the real-time price is obtained endogenously within the model via a supply curve for conventional
generation. These generalisations will not be discussed further
in this paper, because the final result is not very exciting, while
the necessary algebra can become quite complex.
III. Q UANTITY B IDDING : D IFFERENT S HORT AND L ONG
P RICES
In this section, we consider the problem of the optimal
quantity to schedule in the forward market, in the face of a
known forward price and uncertain balancing prices that differ
depending on whether the generator is short or long in the
real-time market. In Section V, we generalise this to bidding
in a forward market in which the forward price is unknown;
in general, this can result in a wind generator submitting a
sloped bid curve rather than a fixed quantity.
1 Throughout, the convention that lower case letters (e.g., w) denote numbers
and capital letters (e.g., W ) denote random variables; the probability density
function for W is fW (w).
3
A. Introduction: GB Market Design
In some power markets the prices felt by generators or
trades in the real-time market differ depending on whether
they are short or long. Examples of this setup include the
British [2], Nordic [3] and Iberian [4] markets. If the wind
generator is short/long in real-time (delivers more/less than
the forward-contracted volume), then this imbalance will be
rectified at the short price πS / long price πL ; these may
respectively be greater/less than the forward market price
πF . This differentiation is justified on the grounds that if a
generator is in imbalance then the System Operator (SO) has
to take actions to make good the imbalance of power, and that
the cost of such actions should then be charged to those who
caused them. It should be emphasised that in many (perhaps
even a majority of) power markets worldwide, wind is not
penalised in this way for being short or long; the discussion
in Section II-B of a single real-time price then applies [1].
1) Great Britain (GB) Market Design:
This section describes how prices are set in the Great
Britain power market; the situation in Scandinavia is very
similar (in terms of divergence of short and long prices in
real-time; see Section II of [3].) The forward market in GB
formally operates by bilateral trading (Chapter 10 of [21]);
there is no centralised pool or power exchange. Trades between
generators and demands are reported to the system operator
(SO) by Gate Closure, which is one hour ahead of real-time.
There are however independent power exchanges, and the SO
publishes a forward price index as a time series for each halfhour settlement period, based on data from these exchanges.
In real-time, the market might either be long (total forwardcontracted generation greater than out-turn demand) or short
(contracted generation less than out-turn demand). Each period’s short price (paid by trades which are short, called the
system buy price in GB) and long price (paid to trades which
are long, called in GB the system sell price) are defined as
follows [22] (forward prices are based on market index data):
•
Market short
– Short price: average price of offers to increase output
accepted by SO.
– Long price: the forward price.
•
Market long
– SBP: the forward price.
– SSP: average price of bids to reduce output accepted
by SO (typically offers will be to pay the SO, as a
reduction in output represents a saving on fuel costs).
Imbalances in the opposite direction to the market are thus
settled at the forward price. The given justification for this is
that such imbalances are helpful to the market, and should
therefore not be penalised relative to the income from a
perfectly balanced contract.
2) Great Britain Historic data:
The historic short, long, and forward market index prices
from 2008 are displayed in Fig. 1. In order to reveal trends
as demand varies, the half hour periods are ranked in order
of increasing demand, and a 101-point moving average is
Fig. 1. Smoothed short and long prices, and forward Market Index Price
(MIP) from the GB market in 2008. The raw price data is taken from [23],
and demand data from [24].
then taken2 . As each time series includes the data from all
half hour periods (not differentiating between hours when the
market was long or short), this graph presents the data as seen
at the time of the forward market when players do not yet
know whether the market will be long or short. As expected,
the overall trend is for all the prices to increase as demand
increases. However, there are clear local maxima in the pricedemand curves at around 72% and 92% of peak demand, and
a less prominent one at 79%.
B. Quantity Bidding: Risk-Neutral Case
1) Model Definition:
If the wind owner is willing to make decisions on the
basis of its expected income, then it is termed risk-neutral.
If it demands a risk premium due to the possibility of its
income being less than the expected value then it would be
termed risk-averse [25]. It is assumed here that the wind
generator is indeed risk-neutral; due to the large number of
time periods over which annual profit is measured, provided
there is no systematic bias in the probability distributions used,
any random fluctuations away from the mean will cancel to a
good approximation.
In this risk-neutral case (and indeed if the wind generator
is assumed averse to quantity risk but neutral to price risk),
it is necessary to specify only expectation values for the realtime prices conditional on wind out-turn, rather than a full
joint probability distribution. Working in terms of expected
values for prices does not sacrifice any generality, save for
these issues around risk-aversion.
The forward price is assumed to be known precisely at the
time the forward contract volume is chosen (bid curves for the
more general case of uncertain forward price will be discussed
in Section V.) The wind owner must therefore decide on its
trading strategy in the forward market, given the uncertainty in
its output in real-time and the real-time prices. The generation
out-turn W and the real-time prices (short price ΠS , and
2 Points are not plotted for the highest and lowest 50 demands, as the moving
average would then be unbalanced.
4
long price ΠL ), conditioned on the information available when
trading in the forward market, are then modelled as random
variables. In GB these expected prices must take into account
that, if the wind owner is in imbalance in the opposite direction
to the market, then in real time the owner pays or is paid the
forward price. It is convenient to define the following expected
short and long penalties
E[∆S |w, πF ] = E[ΠS |w, πF ] − πF
E[∆L |w, πF ] = πF − E[ΠL |w, πF ],
(3)
(4)
where E[Π(S,L) |w, πF ] are the expected real-time prices conditioned on wind out-turn and forward price.
2) Derivation of Optimal Forward Volume:
The expected net revenue, which the risk-neutral wind
owner seeks to maximise, is then
Z q
E[R] = πF q −
E[ΠS |πF , w](q − w)f (w)dw
0
Z
w+
E[ΠL |πF , w](w − q)f (w)dw
(5)
Z q
= πF E[W |πF ] −
E[∆S |πF , w](q − w)f (w)dw
+
q
0
−
Z
q
w
+
E[∆L |πF , w](w − q)f (w)dw,
(6)
where f (w) is the estimated probability density function for
real-time output. This differs from expressions in [6], [7]
in that it does not assume independence between real-time
prices and wind out-turn, and from the equivalent expression
in [3] in that it uses continuous probability distributions rather
than discrete scenarios. This latter difference enables us to
derive closed-form solutions as described below, this type
of derivation not being possible in the scenario approach.
Differentiating, the optimal volume q̂ is
Z
0
q̂
dwE[∆S |πF , w]f (w) =
Z
q̂
w+
dwE[∆L |πF , w]f (w).
(7)
The key features of this expression are as follows:
•
•
In order to evaluate the optimal forward contract volume,
it is sufficient to specify just the expected real-time
prices/penalties, conditioned on the forward price and
wind out-turn. This is much more straightforward than
specifying a full joint distribution for prices and wind
out-turn, or specifying a large set of discrete scenarios.
(7) may be solved using direct numerical methods (numerical integration and equation solving). It is not necessary to use a mathematical optimisation algorithm, which
is required in LP formulations such as [3], [4].
The imbalance prices clearly also depend on the total realtime imbalance volume. The model accounts implicitly for this
dependence through the conditional probability distributions
for prices; this reflects the fact that the market imbalance
volume does not affect the wind-owner’s behaviour directly,
but rather through its effect on real-time prices.
Fig. 2.
Britain.
Expected real-time prices as a function of forward price in Great
IV. A PPLICATION I: U NCORRELATED W IND O UT- TURN
AND R EAL -T IME P RICES
A. Expression for Optimal Forward Contract Volume
If the short and long penalties are independent of the wind
out-turn, then the expression for the optimal forward contract
quantity q̂(πF ) simplifies to
p(W ≤ q̂(πF )) =
E[∆L |πF ]
,
E[∆S |πF ] + E[∆L |πF ]
(8)
where E[∆(S,L) |πF ] is the expected value of ∆(S,L) given
forward price πF . This expression fits with intuition; for
instance, if the expected short price is very high, then the
optimal forward volume is small, reducing exposure to this
high short price.
This expression is similar to those in [6], [7]. The derivation
here makes it explicit that expected penalties given the forward
price should be used, clarifying the statement in [7] that realtime price ‘forecasts or estimates’ must be used.
The formula for the optimal volume is completely closedform, with the benefits of transparency and minimal computational effort that this brings. However, with a substantial
installed wind capacity and consequent large forecast errors
in MW terms, independence of real-time prices and wind outturn might not be a good approximation; in this case the more
generally applicable expressions in Section VI are of greater
relevance.
B. Examples
1) Wind Owner in Great Britain:
Expected real-time prices conditioned on the forward price
for Great Britain in 2008 are displayed in Fig. 2, the data
used being the same as in Fig. 1. These expected RT prices
are calculated by ordering half hour periods by forward price
and taking a 101 point moving average. This carries an
implicit assumption that the expected RT prices depend only
on the forward price, with no seasonal and diurnal effects;
comparison with plots considering one season only shows that
this is reasonable for an illustrative example such as here. A
5
high, then the benefits of the optimal strategy can be very
substantial indeed. This greater potential effect of the short
price is simply due to the expected long price usually being
constrained to values above zero, the most common exception
being when wind is the marginal generator.
V. F ORWARD M ARKET B ID C URVES
A. Uncertain Forward Price in a Pool of Exchange
Fig. 3. Percentage decrease in expected revenue from forward-contracting
expected out-turn, compared to revenue from the optimal strategy presented
here. The out-turn wind load factor conditioned on the information available
at time of forward contracting follows the N (0.5, 0.125) distribution.
full analysis of the historic price data from GB is beyond the
scope of this paper, and will be addressed in future research.
Over a wide range of forward prices the expected realtime, short and long prices are approximated well by 1.2πF
and 0.7πF respectively, giving E[∆S |πF ] ' 0.2πF and
E[∆L |πF ] ' 0.3πF (the highest and lowest forward prices
require a more detailed statistical treatment.) Fig. 2 also
shows the line RT price = forward price (i.e., no penalties);
the vertical distance between that line and the lines 1.2πF
and 0.7πF is equal to the penalty for being short and long
respectively. Over this range, the ratio of the short and long
penalties remain constant, and the optimal contract volume is
given by p(W ≤ q̂) = 0.6. The actual MW optimal volume
depends on the probability distribution for wind out-turn.
This penalty structure has not remained constant over the
years. In the early years of the present market structure, the
short penalty was typically higher. For instance, in 2004, an
equivalent analysis would give an expected short penalty of
approximately 0.4πF , with the expected long penalty again
around 0.3πF .
2) Benefits of Optimal Strategy:
The benefits of following this optimal strategy, instead of
forward-contracting expected out-turn, are demonstrated in
Fig. 3. The wind load factor here is normally distributed with
mean 0.5 and standard deviation (SD) 0.125. This distribution
is consistent with the day-ahead forecast accuracy used in [3],
and is similar to four-hour ahead data from GB. The forecast
SD in GB is, however, about half the size at Gate Closure 1
hour ahead.
The forward price is assumed to be 100 units, with expected
long prices down to a factor of 5 below this, and expected
short prices up to a factor of 5 above. It may be seen that
for small expected penalties (differences between forward and
expected real-time prices), the benefits of the optimal strategy
are not very great. With a moderate short price, optimising
under a very low long price does not provide benefits of more
than a few percent over the alternative of contracting expected
out-turn volume. However, if the expected short price is very
If the forward market price is known with certainty at the
time the bidding strategy is devised, then the above approach
of contracting a fixed volume is sufficient. Also, as seen in (8),
if the ratio of expected short and long penalties is independent
of forward price, then the same forward contract volume will
be optimal irrespective of the forward price.
In general, however, the optimal forward contract volume
can depend on the out-turn forward price (and the resulting
expected penalties). If a bid must be submitted to a pool or
power exchange, then at the time the bid is devised the forward
price cannot be known with certainty; account should be taken
of this when devising the form of the bid. The result is that, in
general, a sloped bid curve is optimal, in which the forward
quantity bid is a function of the forward price. This curve
reflects the opportunity cost of selling in the forward market
rather than the real-time market.
B. Dependence of Optimal Quantity on Forward Price
The optimal bid quantity (8) may be rearranged as
−1
E[∆S |πF ]
.
p(W ≤ q̂(πF )) = 1 +
E[∆L |πF ]
(9)
Thus, for a given forward price, the optimal quantity is determined by the ratio of the expected short and long penalties.
As the forward price increases, the ratio E[∆S |πF ]/E[∆L |πF ]
may increase, stay constant, or decrease. The optimal forward
volume then has the following relationship with that ratio:
• If the ratio increases, then the optimal forward volume
decreases;
• If the ratio is constant, then the optimal forward volume
is constant;
• If the ratio decreases, then the optimal forward volume
increases.
Fig. 4 shows examples for each of these, with E[∆S |πF ] =
0.2πF , and E[ΠS |πF ] = 0.3πF + x where x = 0 (i.e.,
ratio constant), 10 (i.e., ratio increasing) and -10 (i.e., ratio
decreasing).
C. Derivation of Bid Curves
A bid curve for a pool or power exchange may be derived by
swapping the axes on a plot of optimal quantity versus forward
price; the bid curve indicates what price is required for the
generator to supply a given quantity in the forward market. For
the cases where the ratio of expected short to long penalties is
constant with price (i.e., a fixed optimal volume independent
6
a dependence of the short and long penalties on the wind outturn based on the competitive (price-taker) case.
The thermal supply quantity resulting from the expected
wind out-turn is q = d − N µW . Making a quadratic approximation to the thermal supply function about this point, we
obtain:
πT (q)
'
πT (d − N µW ) + c1 (q − (d − N µW ))
+c2 (q − (d − N µW ))2 .
(10)
For a convex thermal supply function, the constants c1 and c2
are positive. Assuming the same form for the dependence of
the short and long prices on the wind out-turn,
E[∆S |πF , w]
Fig. 4. Optimal bid volume as a function of forward price for expected short
price E[∆S |πF ] = 0.2πF and expected long price E[∆L |πF ] = 0.3πF +x,
where x = (−10, 0, 10) /MWh.
of forward price, x = 0 in Fig. 4), or decreasing (implying
optimal volume increasing with forward price, x = −10), the
bid curve will have a typical non-decreasing form.
The case where the ratio of expected short to long penalty
increases as the forward price increases, however, implies a
bid curve where price required decreases as quantity increases
(x = 10). This could not occur where the bid strategy
reflects internal short run marginal costs of supply, as then the
generator would never wish to decrease its supply quantity as
price increases. Indeed, in some markets only non-decreasing
bid curves are permitted, as a protection against exercise of
market power. For the situation here where the forward market
strategy is not based on internal supply costs, the bid curve can
in principle be negative sloping for certain penalty structures,
as demonstrated in (8) and Fig. 4. It should be further noted
that the forward contract volume does not usually affect wind’s
behaviour in real time, when it will generate at the maximum
available output level.
VI. A PPLICATION II: C ORRELATED W IND O UT- TURN
R EAL -T IME P RICES
AND
This section presents two applications of the model developed in Section III-B, in which the wind out-turn and real-time
prices are not independent. Both provide closed-form results
for the optimal bid volume, which give insight into drivers of
the optimal bidding behaviour.
A. Expected Penalties Based on Thermal Supply Function
Suppose that the GB wind owner’s portfolio is welldistributed across the country, so that if its output is w then
the total GB wind output is N w for some constant N . If there
were a single real-time price in a perfectly competitive market,
this would then be πR = πT (d − N w), where πT (q) is the
marginal cost of supply by thermal generation at quantity q.
As described above, the GB market does not have a single
real-time price. However, one can gain insight into how
wind out-turn affects the optimal bid strategy by assuming
E[∆L |πF , w]
= E[∆S |πF , w = µW ] − c1 N (w − µW )
+ c2 N 2 (w − µW )2
(11)
= E[∆L |πF , w = µW ] + c1 N (w − µW )
− c2 N 2 (w − µW )2 .
(12)
For simplicity, this assumes that if the wind generator is well
below (above) its expected real time output then the market
is definitely short (long); for the illustrative example here this
assumption is reasonable at high wind penetrations.
The optimal bid quantity q̂ is then given by
2
E[∆S |πF , W = µW ] − c2 σW
.
E[∆S |πF , W = µW ] + E[∆L |πF , W = µW ]
(13)
Comparison with the expression for uncorrelated real-time
2
prices and wind out-turn (8) shows that the ‘−c2 σW
’ term
acts as a correction to the numerator of the expression for
p(W ≤ q̂), decreasing the optimal forward contract volume
if the thermal supply curve is convex (i.e., positive second
derivative). This fits with intuition, as with a convex thermal
supply curve the real-time prices increase more when wind
is short than they decrease when wind is long. Hence wind
should tend more towards being long to avoid this enhanced
short penalty. This substantial correction term appears despite
the assumption that the wind owner’s behaviour in the forward
market does not affect the forward price.
In competitive markets, it is unusual to assume that one
player knows in advance that all competitors will make the
same decision, and furthermore knows what that decision will
be. However, in this case the wind owners do not really make
a decision regarding their output, rather they all inevitably
generate at the maximum available capacity due to having a
negligible short-run marginal cost of supply in real time. The
2
correction −c2 σW
above follows from an assumption that the
day-ahead error in forecasting the wind owner’s out-turn load
factor follows the same probability distribution as the error in
forecasting the overall GB wind out-turn load factor. However,
in fully realistic cases where this statistical relationship is less
strong, this correction term will be reduced somewhat.
p(W ≤ q̂) =
B. Dependence Structure From a Multivariate Normal Distribution
This example demonstrates a more flexible approach to
modelling the dependence structure, while still providing a
7
direct method for obtaining the optimal forward contract
quantity through numerical solution of an equation.
The wind out-turn is assumed to follow a normal distribution, with probability density function
1
(w − µW )2
√ exp −
fW (w) =
.
(14)
2
2σW
σW 2π
The expected short penalty given wind out-turn w is then
ρS σS
(15)
w − µW .
E[∆S |W = w] = µS +
σW
A similar expression holds for the long penalty in terms of parameters ρL , µL and σL . In a multivariate normal distribution,
these would be the correlation coefficient between wind outturn and prices, and the mean and standard deviation of the
prices, but this assumption of the distribution’s precise form
is not required for the results which follow. The specification
of parameters in (15) must account for the possibility of the
real-time price being set equal to the forward price if the wind
owner is in imbalance in the opposite direction to the market.
Substituting in (7), assuming that the probability of output
near maximum or minimum is negligible, the following equation for the optimal forward contract volume q̂ is obtained:
(µS + µL )p(W ≤ q̂) =
(q̂ − µW )2
(ρL σL + ρS σS )
√
exp −
µL +
(16)
2
2σW
2π
Once again, this may be regarded as adding correction terms
to the ‘uncorrelated’ case (8). It does not provide a completely
closed-form expression for the optimal quantity, but this may
be evaluated by direct numerical solution of the equation. It
is possible to analyse the effects of the relationship between
the wind out-turn and prices as follows:
• If the short penalty is negatively correlated with the wind
out-turn (which is likely to be the case) then, compared to
the independent case, the wind owner reduces the forward
contract volume at a given price. This behaviour mitigates
the consequences of an increased short price when the
wind out-turn is low.
• If the long penalty is positively correlated with the wind
out-turn (which is likely to be the case) then, compared
to the independent case, the wind owner increases the
forward contract volume at a given price. This behaviour
mitigates the consequences of a decreased long price
when the wind out-turn is high.
As a result, the behaviour of wind relative to the independent
case is determined by the relative volatilities of the short
and long prices, as well as by the degree to which they are
correlated with the wind out-turn.
VII. R ISK -AVERSE T RADING B EHAVIOUR
A. VaR and CVaR
The previous results have all been for risk-neutral behaviour,
where the wind owner seeks to maximise expected revenue,
without any consideration of variance of revenues or of how far
the revenue can decrease below that expected value. We now
relax that assumption and consider risk-averse behaviour. The
most common measures used to analyse risk-averse trading
strategies are Value at Risk (VaR) and Conditional Value at
Risk (CVaR) [26].
The VaR for the revenue R at the α confidence level is
defined as the revenue rα such that p(R ≤ rα ) = 1 − α.
The CVaR for the revenue R at the α confidence level is then
defined as the expected revenue conditioned on the revenue
being below rα , i.e., in mathematical notation the CVaR is
E[R|R ≤ rα ].
CVaR is more commonly used as the objective function
when optimising risk-averse strategies, because unlike VaR
it exhibits appropriate mathematical behaviour under general
conditions (‘coherence’) [26], and can be included within LP
formulations [27]. CVaR also allows a smooth transition from
risk-neutral behaviour (i.e., α = 0) to increasing degrees of
risk aversion by varying the parameter α.
B. Risk-averse Forward Trading Strategy
Our approach of deriving closed-form expressions may be
extended to risk-averse behaviour, provided that the long and
short real-time prices can be treated using expected values.
The wind owner would then be assumed to be averse to
quantity risk but not price risk. Aversion to price risk requires
evaluation of various double integrals of the joint probability
distribution for prices and wind out-turn. While this full
risk-averse problem is numerically tractable, the degree of
transparency seen in other results from this paper is not
available.
Given the large number of time periods over which a trading
strategy will be used, in practice the degree of risk aversion
that wind owners demonstrate in any single period may be
limited; in the long run, a large degree of cancellation is to
be expected between the consequences of below-average and
above-average wind out-turns for single periods. A case might
be made that aversion to price-risk is equally important as
quantity risk, due to the possibility of extreme spikes in the
short price. Given that the GB market as a whole is typically
long, further investigation of whether (and in what way) price
risk aversion is exhibited would be valuable. However, the
greatest long-run risk could well be systematic error in the
estimated wind distribution, so that the calculated optimal
quantity has a bias towards being either above or below the
true optimum. In this case, quantity risk aversion could be
more important.
Based on this limited picture of aversion to wind out-turn
risk, assuming a known forward price πF and real-time prices
independent of wind out-turn, the CVaR at the α confidence
level is
Z
E[ΠS |πF ] q
dw(q − w)f (w)
CVaRα = πF q −
1−α
Z wα 0
E[ΠL |πF ]
dw(w − q)f (w), (17)
+
1−α
q
where p(W ≤ wα ) = 1−α. It is also assumed that the optimal
forward quantity is less than wα (which is reasonable for small
α). Differentiating to obtain the optimal bid volume q̂:
E[∆L |πF ]
.
(18)
p(W ≤ q̂) = (1 − α)
E[∆S |πF ] + E[∆L |πF ]
8
averse to production volume risk. Examples based on data
from Great Britain have been presented. The methods are also
directly applicable to other markets such as those described in
[3], [4].
The new methods presented here require specification only
of the expected real-time prices given the forward price and
wind out-turn; this is simpler than specifying a full joint
distribution or a representative finite set of scenarios. The
new analytical approach is also much more computationally efficient than previous linear programming formulations.
Moreover, in the risk-neutral case the new approach does not
sacrifice any generality in the optimisation problem for a single
period. When the forward price is unknown, the result is in
general a sloped bid curve, reflecting the opportunity cost of
trading in the forward as opposed to real-time market.
Fig. 5. Expected revenue, and 95%-VaR, for varying risk aversion parameter
α. The revenue is measured in units of [forward price] multiplied by [installed
wind capacity].
The optimal forward contracted volume decreases as the
degree of risk aversion increases. This is as anticipated since,
under risk aversion, the increased expected utility resulting
from long payments does not compensate for the loss of
expected utility resulting from penalties for being short. That
is, poorer outcomes are weighted more highly than good
outcomes under risk-aversion.
C. Example
This section presents an illustrative example, with parameters based on wind and price data from GB. The short and
long prices are assumed to be respectively 120% and 70% of
the forward price, based on the data in Fig. 2. The out-turn
wind load factor is assumed to have a normal distribution with
mean 0.42 and standard deviation 0.12. These parameters are
based on the variation in wind out-turn for those hours in 2008
during which the 4 hour ahead load factor was between 40%
and 50%.
Results using the wind-volume-risk-averse strategy (18) are
shown in Fig. 5. When the risk aversion parameter α is below
50%, a small decrease (less than 2%) in the expected revenue
is seen. As a measure of the positive effect of the risk-averse
strategy, VaR at the 95% confidence level is also plotted (VaR
is used instead of CVaR here because of its more intuitive
interpretation.) Over the same range of α, the gain in the
95%-VaR is slightly larger in absolute terms (and considerably
larger in relative terms) than the decrease in expected revenue;
a similar result has been noted in [4]. However, as discussed
in Section VII-B, it is doubtful whether looking at individual
periods in isolation is the correct approach to considering risk
aversion in this application.
VIII. C ONCLUSIONS
We have presented new analytical expressions for determining the optimal forward market strategy for wind generators,
including cases where the real-time prices and real-time production are not independent, and where the wind owner is
ACKNOWLEDGMENT
We acknowledge valuable discussions with R. Green, D.
Newberry, M. Roberts, their partners in the Supergen-Flexnet
consortium, and the Probability and Stochastic Models group
at Heriot-Watt University. We express particular thanks to
National Grid for sharing historic data on Great Britain wind
generation.
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Chris Dent (M’08) is a Research Fellow in the
School of Engineering and Computing Sciences,
Durham University, UK. He obtained his BA in
Mathematics from Cambridge University in 1997,
Ph.D. in Theoretical Physics from Loughborough
University in 2001, and MSc in Operational Research from the University of Edinburgh in 2006.
From 2007-2009 he was with the University of
Edinburgh. His research interests lie in power system
optimisation, risk modelling, economics and renewables integration.
Dr. Dent is a Chartered Physicist and an Associate of the Operational
Research Society, and a member of the IET and Cigré.
Janusz W. Bialek (SM’08) is Professor of Electrical
Power and Control at Durham University, UK. He
holds M.Eng (1977) and Ph.D. (1981) degrees from
Warsaw University of Technology, Poland. From
1981-1989 he was with Warsaw University of Technology, from 1989-2002 with Durham University,
UK, and from 2003-2009 he held the Chair of Electrical Engineering at the University of Edinburgh.
He has co-authored 2 books and over 100 technical
papers. His research interests include power system
economics, power system dynamics and sustainable
energy systems.
Benjamin F. Hobbs (SM’01, F’08) received the
Ph.D. degree in environmental system engineering
from Cornell University, Ithaca, NY, in 1983. He is
Schad Professor of Environmental Management at
the Johns Hopkins University, Baltimore, MD, where
he also directs the JHU Environment, Sustainability
& Health Institute. In 2009-2010 he was on sabbatical with the Electricity Policy Research Group at the
University of Cambridge. Dr. Hobbs is a member of
the California ISO Market Surveillance Committee.